On the Dynamics of Influence and Appraisal Networks

Francesco Bullo

Department of Mechanical Engineering

Center for Control, Dynamical Systems & Computation

University of California at Santa Barbara

http://motion.me.ucsb.edu

30th Chinese Control and Decision ConferenceShenyang, China, June 9, 2018

Acknowledgments

Peng JiaDiscover Financial

Ana MirTabatabaeiApple

Wenjun MeiETH

Xiaoming DuanUCSB

Noah E. FriekinUCSB

Kyle LewisUCSB

Ge ChenChinese Academy of

Sciences

New text “Lectures on Network Systems”

Lectures onNetwork Systems

Francesco Bullo

With contributions byJorge Cortés

Florian DörflerSonia Martínez

Lectures on Network SystemsFrancesco Bullo

These lecture notes provide a mathematical introduction to multi-agent

dynamical systems, including their analysis via algebraic graph theory

and their application to engineering design problems. The focus is on

fundamental dynamical phenomena over interconnected network

systems, including consensus and disagreement in averaging systems,

stable equilibria in compartmental flow networks, and synchronization

in coupled oscillators and networked control systems. The theoretical

results are complemented by numerous examples arising from the

analysis of physical and natural systems and from the design of

network estimation, control, and optimization systems.

Francesco Bullo is professor of Mechanical Engineering and member

of the Center for Control, Dynamical Systems, and Computation at the

University of California at Santa Barbara. His research focuses on

modeling, dynamics and control of multi-agent network systems, with

applications to robotic coordination, energy systems, and social

networks. He is an award-winning mentor and teacher.

Francesco BulloLectures on N

etwork System

s

Lectures on Network Systems, 1 editionISBN 978-1-986425-64-3

For students: free PDF for downloadFor instructors: slides and answer keyshttp://motion.me.ucsb.edu/book-lnshttps://www.amazon.com/dp/1986425649

300 pages (plus 200 pages solution manual)3K downloads since Jun 2016150 exercises with solutions

Linear Systems:

1 social, sensor, robotic & compartmental examples,

2 matrix and graph theory, with an emphasis onPerron–Frobenius theory and algebraic graph theory,

3 averaging algorithms in discrete and continuous time,described by static and time-varying matrices, and

4 positive & compartmental systems, dynamical flowsystems, Metzler matrices.

Nonlinear Systems:

5 nonlinear consensus models,

6 population dynamic models in multi-species systems,

7 coupled oscillators, with an emphasis on theKuramoto model and models of power networks

Dynamics and learning in social systems

Dynamic phenomena on dynamic social networks

1 opinion formation, information propagation, collective learning,task decomposition/allocation/execution

2 interpersonal network structures, e.g., influences & appraisals

Questions on collective intelligence, rationality & performance:

wisdom of crowds vs. group think

influence centrality (democracy versus autocracy)

collective learning or lack thereof

Dynamics and learning in social systemsMathematical sociology + systems/controls

opinion dynamics over influence networks

seminal works: French ’56, Harary ’59, DeGroot ’74, Friedkin ’90

recently: bounded confidence, learning, social power

key object: row stochastic matrix

dynamics of appraisal networks and balance theory

seminal works: Heider ’46, Cartwright ’56, Davis/Leinhardt ’72

recently: dynamic balance, empirical studies

key object: signed matrix

Not considered today:

other dynamic phenomena (epidemics)

static network science (clustering)

game theory and strategic behavior (network formation)

Selected literature on math sociology and systems/control

F. Harary, R. Z. Norman, and D. Cartwright. Structural Models: An Introduction tothe Theory of Directed Graphs.

John Wiley & Sons, 1965.

ISBN 047135130X (Institute for Social Research, University of Michigan)

M. O. Jackson. Social and Economic Networks.

Princeton University Press, 2010.

ISBN 0691148201

D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About aHighly Connected World.

Cambridge University Press, 2010.

ISBN 0521195330

N. E. Friedkin and E. C. Johnsen. Social Influence Network Theory: A SociologicalExamination of Small Group Dynamics.

Cambridge University Press, 2011.

ISBN 9781107002463

exploding literature on social networks from sociology, physics, CS/engineering

Selected literature on opinion dynamics

J. R. P. French. A formal theory of social power.

Psychological Review, 63(3):181–194, 1956.

doi:10.1037/h0046123

M. H. DeGroot. Reaching a consensus.

Journal of the American Statistical Association, 69(345):118–121, 1974.

doi:10.1080/01621459.1974.10480137

N. E. Friedkin and E. C. Johnsen. Social influence and opinions.

Journal of Mathematical Sociology, 15(3-4):193–206, 1990.

doi:10.1080/0022250X.1990.9990069

A. V. Proskurnikov and R. Tempo. A tutorial on modeling and analysis of dynamicsocial networks. Part I.

Annual Reviews in Control, 43:65–79, 2017.

doi:10.1016/j.arcontrol.2017.03.002

Selected literature on social power & reflected appraisal

C. H. Cooley. Human Nature and the Social Order.

Charles Scribner Sons, New York, 1902

V. Gecas and M. L. Schwalbe. Beyond the looking-glass self: Social structure andefficacy-based self-esteem.

Social Psychology Quarterly, 46(2):77–88, 1983.

URL http://www.jstor.org/stable/3033844

N. E. Friedkin. A formal theory of reflected appraisals in the evolution of power.

Administrative Science Quarterly, 56(4):501–529, 2011.

doi:10.1177/0001839212441349

Outline

1

Influence systems: statistical results on empirical data

N. E. Friedkin, P. Jia, and F. Bullo. A theory of the evolution ofsocial power: Natural trajectories of interpersonal influence systemsalong issue sequences.Sociological Science, 3:444–472, 2016.doi:10.15195/v3.a20

N. E. Friedkin and F. Bullo. How truth wins in opinion dynamicsalong issue sequences.Proceedings of the National Academy of Sciences, 114(43):11380–11385, 2017.doi:10.1073/pnas.1710603114

2 Influence systems: the mathematics of social power

3 Appraisal systems and collective learning

Opinion dynamics and social power along sequences

Deliberative groups in social organization

government: juries, panels, committees

corporations: board of directors

universities: faculty meetings

Natural social processes along sequences

opinion dynamics for single issue?

levels of openness and closure along sequence?

influence accorded to others? emergence of leaders?

Groupthink = “deterioration of mental efficiency . . . fromin-group pressures,” by I. Janis, 1972

Wisdom of crowds = “group aggregation of information resultsin better decisions than individual’s” by J. Surowiecki, 2005

Postulated mechanisms for opinion dynamics 1/2

French-DeGroot averaging model

y+i := average(yi , {yj , j is neighbor of i}

)

y(k + 1) = Ay(k)

where A is nonnegative and row-stochasticConsensus under mild connectivity assumptions:

limk→∞

y(k) = (c>y(0)) 1n

self-weight = level of closure: aii diagonal entries of influence matrixsocial power: ci entries of dominant left eigenvector c = vleft(A)

Postulated mechanisms for opinion dynamics 2/2

Averaging (French-DeGroot model)

y(k + 1) = Ay(k) limk→∞ y(k) = (c>y(0))1n

Averaging + attachment to initial opinion (F-J model)

y(k + 1) = (In − Λ)Ay(k) + Λy(0),

Λ = diag(A)

Convergence under mild connectivity+stubburness assumptions:

limk→∞

y(k) = V · y(0), for V = (In − (In − Λ)A)−1Λ

c = V>1n/n = average contribution of each agent

self-weight = level of closure: aii diagonal entries of influence matrixsocial power: ci entries of centrality vector

Today we skip these proofs

Mathematical analysis of French-DeGroot and F-J models is wellunderstood:

Jordan normal form

Perron-Frobenius theory

algebraic graph theory (connectivity, periodicity, etc)

Experiments on opinion formation and influence networksdomains: risk/reward choice, analytical reliability, resource allocation

30 groups of 4 subjects in a face-to-face discussion

sequence of 15 issues

each issue is risk/reward choice:

what is your minimum level of confidence (scored 0-100)required to accept a risky option with a high payoff ratherthan a less risky option with a low payoff?e.g.: medical, financial, professional, etc

“please, reach consensus” pressure

On each issue, each subject recorded (privately/chronologically):1 an initial opinion prior to the-group discussion,2 a final opinion after the group-discussion (3-27 mins),3 an allocation of “100 influence units”

(“these allocations represent your appraisal of the relative influence ofeach group member’s opinion on yours”).

(1/3) Prediction of individual final opinions

Balanced random-intercept multilevel longitudinal regression

(a) (b) (c)

F-J prediction 0.897∗∗∗ 1.157∗∗∗

(0.018) (0.032)

initial opinions −0.282∗∗∗

(0.031)

log likelihood -8579.835 -7329.003 -7241.097

Standard errors are in parentheses; ∗∗ p ≤ 0.01, ∗∗∗ p ≤ 0.001; maximum

likelihood estimation with robust standard errors; n = 1, 800.

FJ averaging model is predictive for risk/reward choice issues

Extensions to: intellective and resource allocation issues

Risk/reward choice: N. E. Friedkin, P. Jia, and F. Bullo. A theory of theevolution of social power: Natural trajectories of interpersonal influencesystems along issue sequences.

Sociological Science, 3:444–472, 2016.

doi:10.15195/v3.a20

Intellective issueTwo medical teams are working independently to achieve a cure for a disease.

Team A succeeds ifproblems A1 and A2 with P[A1] = 0.60 and P[A2] = 0.45.

Team B succeeds ifproblems B1, B2, and B3, with P[B1] = 0.80, P[B2] = 0.85, P[B3] = 0.95

What is your estimate of the probability that the disease will be cured?

Multidimensional resource allocationDiet problem: Given 4 food groups: Fruits, Vegetables, Grains, and Meats.

What do you recommend as min and max percent of food consumptionin terms of (1) Fruits or Vegetables, (2) Grains, and (3) Meats?

What are your ideal percentages in your preferred min/max ranges?

Recent empirical and theoretical results

Averaging models are predictive

N. E. Friedkin, P. Jia, and F. Bullo. A theory of the evolution of social power:Natural trajectories of interpersonal influence systems along issue sequences.

Sociological Science, 3:444–472, 2016.

doi:10.15195/v3.a20

N. E. Friedkin and F. Bullo. How truth wins in opinion dynamics along issuesequences.

Proceedings of the National Academy of Sciences, 114(43):11380–11385, 2017.

doi:10.1073/pnas.1710603114

Empirical evidence that (1) FJ model substantially clarifies how truthwins in groups engaged in sequences of intellective issues (2) learningand reflected appraisal take place

N. E. Friedkin, W. Mei, A. V. Proskurnikov, and F. Bullo. Mathematical structuresin group decision-making on resource allocation distributions.

Submitted

Empirical evidence that (1) FJ model provides quantitative mechanisticexplanation for uncertain multi-objective decision making problem and(2) FJ provides detailed explanation for group satisficing solutions

Opinion dynamics along sequencesPostulated mechanism for network evolution

From Wikipedia

1. Reflected appraisal = a person’s perception of how others see andevaluate him or her.

2. This process has been deemed important to the development of aperson’s self-esteem, because it includes interaction with people outsideoneself.

3. The reflected appraisal process concludes that people come to thinkof themselves in the way they believe others think of them.

Reflected appraisal process (Cooley 1902 and Friedkin 2011)

Along issues s = 1, 2, . . . , individual dampens/elevatesself-weight according to prior influence centrality

self-weights := relative control on prior issues = social power

(2/3) Prediction of individual level of closure

Balanced random-intercept multilevel longitudinal regression

individual’s “closure to influence” as predicted by:

individual’s prior centrality ci (s)

individual’s time-averaged centrality ci (s) = 1s

∑st=1 ci (t)

(a) (b) (c)

ci (s) 0.336∗∗∗

ci (s) 0.404∗∗

s 0.002 −0.018∗∗∗

s × ci (s) 0.171s × ci (s) 0.095∗∗∗

log likelihood -367.331 -327.051 -293.656

prior and cumulative prior centrality predicts individual closure

(3/3) Prediction of cumulative influence centrality

complete closure to influence on issue s + 1 of the issue sequence increases with the individual’sprior time-averaged influence centrality Ti(s). Figure 7 shows that the frequency of instances ofgroup members who are completely closed to influence is elevated along the issue sequence. Inother words, the stabilizing relative di↵erences of individuals’ Ti(s) centralities become increasinglyindicative of the unequal rates at which individuals are accumulating centrality. Hence, the findingin Tables 2 and 3 on the increasing e↵ect of Ti(s) along the sequence.

Figure 5: Evolution of individuals’ cumulative influence centralityPs

t=1 Ci(t) and time-averagecentrality Ti(s) = 1

s

Pst=1 Ci(t) for each individual in each of the 30 groups along the issue sequence.

02

46

02

46

02

46

02

46

02

46

0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

Cum

ulat

ive

influ

ence

cen

tralit

y

Issue sequence

0.2

.4.6

.80

.2.4

.6.8

0.2

.4.6

.80

.2.4

.6.8

0.2

.4.6

.8

0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15

1 2 3 4 5 6

7 8 9 10 11 12

13 14 15 16 17 18

19 20 21 22 23 24

25 26 27 28 29 30

Cum

ulat

ive

influ

ence

cen

tralit

y

Issue sequenceTime averaged

Figure 6: Prior time-averaged centrality Ti(s) = 1s

Pst=1 Ci(t) of individual i and the individual’s

probability of complete closure to influence aii = 1 � wii = 0 on issue s + 1. Balanced logisticrandom-intercept multilevel longitudinal design. Odds-ratio estimates: �0 = 0.063 (s.e. = 0.018),p 0.001; �1 = 54.798 (s.e. = 45.018), p 0.001. The vertical line indicates the maximumobserved value of Ti(s) in the dataset.

0.2

.4.6

.8Pr

obab

ility

of c

ompl

ete

clos

ure

to in

fluen

ce o

n is

sue

s+1

0 .2 .4 .6 .8 1An individual's time-averaged cummulative centrality on issue s

15

individuals accumulate influence centralities at different rates,and their time-average centrality stabilizes to constant values

Outline

1 Influence systems: statistical results on empirical data

2

Influence systems: the mathematics of social power

P. Jia, A. MirTabatabaei, N. E. Friedkin, and F. Bullo. Opiniondynamics and the evolution of social power in influence networks.SIAM Review, 57(3):367–397, 2015.doi:10.1137/130913250

G. Chen, X. Duan, N. E. Friedkin, and F. Bullo. Social powerdynamics over switching and stochastic influence networks.IEEE Transactions on Automatic Control, May 2017.doi:10.1109/TAC.2018.2822182

3 Appraisal systems and collective learning

Opinion dynamics and social power along issue sequences

French-DeGroot averaging model

y(k + 1) = Ay(k)

Consensus under mild assumptions:

limk→∞

y(k) = (vleft(A) · y(0))1n

where vleft(A) is social power

Aii =: xi are self-weights / self-appraisal = level of closure

let Wij be relative interpersonal accorded weightsdefine Aij =: (1− xi )Wij so that

A(x) = diag(x) + diag(1n − x)W

vleft(W ) = (w1, . . . ,wn) = dominant eigenvector for W

Opinion dynamics and social power along issue sequences

Reflected appraisal phenomenon (Cooley 1902 and Friedkin 2011)

along issues s = 1, 2, . . . , individual dampens/elevatesself-weight according to prior influence centrality

self-weights relative control on prior issues = social power

self-appraisal

reflected appraisal mechanism

x(s + 1) = vleft(A(x(s)))

x(s) A(x(s)) vleft(A(x(s)))

influence network social power

Dynamics of the influence network

Existence and stability of equilibria?Role of network structure and parameters?Emergence of autocracy and democracy?

Theorem: For strongly connected W and non-trivial initial conditions

1 unique fixed point x∗ = x∗(w1, . . . ,wn)

2 convergence = forgets initial condition

lims→∞

x(s) = lims→∞

vleft(A(x(s))) = x∗

3 accumulation of social power and self-appraisalfixed point x∗ has same ordering of (w1, . . . ,wn)x∗ is an extreme version of (w1, . . . ,wn)

Emergence of democracy

If W is doubly-stochastic:

1 the non-trivial fixed point is1n

n

2 lims→∞

x(s) = lims→∞

vleft(A(x(s))) =1n

n

Uniform social power

No power accumulation = evolution to democracy

issue 1 issue 2 issue 3 . . . issue N

Emergence of autocracy

If W has star topology with center j :

1 there are no non-trivial fixed points

2 lims→∞

x(s) = lims→∞

vleft(A(x(s))) = ej

Autocrat appears in center node of star topology

Extreme power accumulation = evolution to autocracy

issue 1 issue 2 issue 3 . . . issue N

Analysis methods

1 existence of x∗ viaBrower fixed point theorem

2 monotonicity:imax and imin are forward-invariant

imax = argmaxjxj(0)

x∗j

=⇒ imax = argmaxjxj(s)

x∗j, for all subsequent s

3 convergence via variation on classic “max-min” Lyapunov function:

V (x) = maxj

(ln

xjx∗j

)−min

j

(ln

xjx∗j

)strictly decreasing for x 6= x∗

Stochastic models with cumulative memoryOther extensions: modified models, reducible W , periodic W ...

1 assume noisy interpersonal weights W (s) = W0 + N(s)assume noisy perception of social powerx(s + 1) = vleft(A(x(s))) + n(s)Thm: practical stability of x∗

0

0 0

0.2

0.20.2

0.4

0.40.4

0.6

0.60.6

0.8

0.80.8

1

11

0

0 0

0.2

0.20.2

0.4

0.40.4

0.6

0.60.6

0.8

0.80.8

1

11

0

0 0

0.2

0.20.2

0.4

0.40.4

0.6

0.60.6

0.8

0.80.8

1

11

0

0 0

0.2

0.20.2

0.4

0.40.4

0.6

0.60.6

0.8

0.80.8

1

11

2 assume self-weight := cumulative average of prior social power

x(s + 1) = (1− α(s))x(s) + α(s)(vleft(A(x(s))) + n(s)

)

Thm: a.s. convergence to x∗ (under technical conditions)

Recent extensions on social power evolution

X. Chen, J. Liu, M.-A. Belabbas, Z. Xu, and T. Basar. Distributed evaluation andconvergence of self-appraisals in social networks.

IEEE Transactions on Automatic Control, 62(1):291–304, 2017.

doi:10.1109/TAC.2016.2554280

M. Ye, J. Liu, B. D. O. Anderson, C. Yu, and T. Basar. On the analysis of theDeGroot-Friedkin model with dynamic relative interaction matrices.

In IFAC World Congress, pages 11902–11907, Toulouse, France, July 2017.

doi:10.1016/j.ifacol.2017.08.1426

P. Jia, N. E. Friedkin, and F. Bullo. Opinion dynamics and social power evolutionover reducible influence networks.

SIAM Journal on Control and Optimization, 55(2):1280–1301, 2017.

doi:10.1137/16M1065677

Z. Askarzadeh, R. Fu, A. Halder, Y. Chen, and T. T. Georgiou. Stability theory in`1 for nonlinear Markov chains and stochastic models for opinion dynamics, 2017.

URL https://arxiv.org/pdf/1706.03158

Summary (Social Influence)

New perspective on influence networks and social power

designed/executed/analyzed experiments on group discussions

proposed/analyzed/validated dynamical models with feedback

novel mechanism for power accumulation / emergence of autocracy

Open directions

model robustness

dynamics of interpersonal appraisals

larger-scale online experiments

intervention strategies for optimal group discussions

No one speaks twice, until everyone speaks onceRobert’s Rules of Order & parliamentary procedures

Outline

1 Influence systems: statistical results on empirical data

2 Influence systems: the mathematics of social power

3

Appraisal systems and collective learning

W. Mei, N. E. Friedkin, K. Lewis, and F. Bullo. Dynamic models ofappraisal networks explaining collective learning.IEEE Transactions on Automatic Control, 2018.doi:10.1109/TAC.2017.2775963

Appraisal systems and collective learning

Teams and tasks

individuals with skills

executing a sequence of tasks

related through networks of interpersonal appraisals and influence

Natural social processes along sequences

how is task decomposed, assigned and executed?

how do individuals learn about each other?

how does group performance evolve?

models/conditions for learning correct appraisals andachieving optimal assignments

model/conditions for failure to learn and correctly assign

Selected literature on learning in appraisal systems

D. M. Wegner. Transactive memory: A contemporary analysis of the group mind.

In B. Mullen and G. R. Goethals, editors, Theories of Group Behavior, pages185–208. Springer, 1987.

doi:10.1007/978-1-4612-4634-3

K. Lewis. Measuring transactive memory systems in the field: Scale developmentand validation.

Journal of Applied Psychology, 88(4):587–604, 2003.

doi:10.1037/0021-9010.88.4.587

J. R. Austin. Transactive memory in organizational groups: the effects of content,consensus, specialization, and accuracy on group performance.

Journal of Applied Psychology, 88(5):866, 2003.

doi:10.1037/0021-9010.88.5.866

Collective Learning Model

Transactive memory system (TMS)

collective knowledge on who knows what

task execution & observation lead groupto increasingly accurate knowldge & consensus

1 n individuals with skill levels x ∈ ∆n

2 TMS = appraisal matrix row stochastic

3 workload assignment w ∈ ∆n

4 optimal assignment: w∗ = x

xw

A

xw

AA

Collective Learning Model: Key Assumptions

assignment rulebased on appraisal matrix A(t)

task executionbased on skills x

appraise/influence dynamics

assignment w(t)

individual relativeperformance

task t

appraisalmatrix A(t)

Key assumptions of assign/appraise dynamics

1 Static assignment by appraisal centrality or appraisal average

2 Static individual performance pi = xi/wi

3 Observation of own vs average performance

φi = own performance − average of observed subgroup performance

4 appraise/influence: elevate/dampen self-appraisal + opinion exchange

Collective Learning Model: Result 1/3

1. TMS is akin to a manager

Along assign/appraise dynamics: generalized replicator equation

wi = wi

(aiiφi −

∑

k

akkwkφk

)

akin to replicator equation modeling a “manager dynamics”

wi = wi

(pi −

∑

k

wkpk

)=⇒ w(t)→ x

Collective Learning Model: Result 2/3

1. TMS is akin to a manager

2. Opinion exchange compensates for lack of observation.

Conditions for asymptotic optimization

assign/appraise dynamics:

strongly connected observation network ⇒ w(t)→ w∗ = x

additionally with influence dynamics:

globally reachable node in observation network ⇒ w(t)→ w∗ = x

moreover: consensus on correct appraisals

assign/appraise dynamics:xw

A

xw

A

xw

A

xw

A

assign/appraise + influence dynamics:xw

A

xw

A

xw

A

xw

A

Collective Learning Model: Result 3/3

1. TMS is akin to a manager

2. Opinion exchange compensates for lack of observation.

3. Causes of incorrect learning and suboptimal assignment:

assignment rule: appraisal average (and no influence dynamics)

appraise dynamics: observation graph without connectivity properties

influence dynamics: prejudice model (F-J model)

Summary

Contributions

dynamics and feedback in sociology and organization science

a new perspective on social power, self-appraisal, influence networks

a new explanation of team learning and rationality

self-appraisal

reflected appraisal mechanism

x(s + 1) = vleft(A(x(s)))

x(s) A(x(s)) vleft(A(x(s)))

influence network social power

assignment rulebased on appraisal matrix A(t)

task executionbased on skills x

appraise/influence dynamics

assignment w(t)

individual relativeperformance

task t

appraisalmatrix A(t)

Next steps

1 theoretical analysis of increasingly realistic models

2 validation via human subject experiments on larger networks

3 Outreach/collaboration for control community withsociologists, psychologists, organization scientistson problems from Mathematical Sociology and Network Science